All of the 5th grade teachers and students from Santa Rita went on a field trip to an archaeology museum. Tickets were $$5.00$ each for teachers and $$3.50$ each for students, and the group paid $$50.00$ in total. The next month, the same group visited a natural history museum where the tickets cost $$15.00$ each for teachers and $$12.50$ each for students, and the group paid $$170.00$ in total. Find the number of teachers and students on the field trips.
Answer: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5x+3.5y = 50}$ ${15x+12.5y = 170}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-15x-10.5y = -150}$ ${15x+12.5y = 170}$ Add the top and bottom equations together. $ 2y = 20 $ $ y = \dfrac{20}{2}$ ${y = 10}$ Now that you know ${y = 10}$ , plug it back into $ {5x+3.5y = 50}$ to find $x$ ${5x + 3.5}{(10)}{= 50}$ $5x+35 = 50$ $5x = 15$ $x = \dfrac{15}{5}$ ${x = 3}$ You can also plug ${y = 10}$ into $ {15x+12.5y = 170}$ and get the same answer for $x$ ${15x + 12.5}{(10)}{= 170}$ ${x = 3}$ There were $3$ teachers and $10$ students on the field trips.